Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 1/23/25 3:32 AM, WM wrote:

On 22.01.2025 13:10, Richard Damon wrote:

On 1/22/25 5:36 AM, WM wrote:

On 22.01.2025 00:41, Richard Damon wrote:

On 1/21/25 7:48 AM, WM wrote:

On 21.01.2025 13:17, Richard Damon wrote:
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the matching happens as a collective whole. It can't complete if you don't move from the individual step to the collective step.

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So it is.
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And thus,

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there are visible and dark numbers. As you said: We have to move from the individual step to the collective step.
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But the numbers being used didn't change, only the logic.

 Numbers do not change. Only their state of being known.

So, why do you claim they changed?
You seem to think that you can't use "visible" numbers collectively, they become dark when you do, nor can your "dark" numbers be used individually,

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I guess when we talk about the set { 0, 1, 2, 3, ... } you must be defining "4" as a "dark number" as that was the point we shifted from listing them individually to collectively.

 No. Every number that is defined in a system by its FISON is visible. Many numbers smaller than 10^99 are defined on the pocket calculator. No greater number can be defined in that system.

Numbers are not definied by its "FISON", its FISON is defined by the number.
ALL Natural Numbers are defined, and thus visible, and not "dark"
There are an infinite set of FISONs, one for every Natural Number.

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The basic problem with your "dark numbers" is that no number itself is "dark", the collective nature is just how we chose to use that number.

 It is how we *can* chose them.

And what number that exists can't we choose, so it becomes dark?
Your logic is just built of lies and falsehoods.
Your "Darkness" is just a result of your ignorance, like "darkest Africa" was the area that was unknown.
Your "Dark Numbers" are just the numbers your logic can not handle.

 Regards, WM